Boltzmann probabilities

  • Energy Entropy Boltzmann probabilities
    • assignment Gibbs sum for a two level system

      assignment Homework

      Gibbs sum for a two level system
      Gibbs sum Microstate Thermal average energy Thermal and Statistical Physics 2020
      1. Consider a system that may be unoccupied with energy zero, or occupied by one particle in either of two states, one of energy zero and one of energy \(\varepsilon\). Find the Gibbs sum for this system is in terms of the activity \(\lambda\equiv e^{\beta\mu}\). Note that the system can hold a maximum of one particle.

      2. Solve for the thermal average occupancy of the system in terms of \(\lambda\).

      3. Show that the thermal average occupancy of the state at energy \(\varepsilon\) is \begin{align} \langle N(\varepsilon)\rangle = \frac{\lambda e^{-\frac{\varepsilon}{kT}}}{\mathcal{Z}} \end{align}

      4. Find an expression for the thermal average energy of the system.

      5. Allow the possibility that the orbitals at \(0\) and at \(\varepsilon\) may each be occupied each by one particle at the same time; Show that \begin{align} \mathcal{Z} &= 1 + \lambda + \lambda e^{-\frac{\varepsilon}{kT}} + \lambda^2 e^{-\frac{\varepsilon}{kT}} \\ &= (1+\lambda)\left(1+e^{-\frac{\varepsilon}{kT}}\right) \end{align} Because \(\mathcal{Z}\) can be factored as shown, we have in effect two independent systems.

    • assignment Pressure and entropy of a degenerate Fermi gas

      assignment Homework

      Pressure and entropy of a degenerate Fermi gas
      Fermi gas Pressure Entropy Thermal and Statistical Physics 2020
      1. Show that a Fermi electron gas in the ground state exerts a pressure \begin{align} p = \frac{\left(3\pi^2\right)^{\frac23}}{5} \frac{\hbar^2}{m}\left(\frac{N}{V}\right)^{\frac53} \end{align} In a uniform decrease of the volume of a cube every orbital has its energy raised: The energy of each orbital is proportional to \(\frac1{L^2}\) or to \(\frac1{V^{\frac23}}\).

      2. Find an expression for the entropy of a Fermi electron gas in the region \(kT\ll \varepsilon_F\). Notice that \(S\rightarrow 0\) as \(T\rightarrow 0\).

    • assignment Free energy of a two state system

      assignment Homework

      Free energy of a two state system
      Helmholtz free energy entropy statistical mechanics Thermal and Statistical Physics 2020
      1. Find an expression for the free energy as a function of \(T\) of a system with two states, one at energy 0 and one at energy \(\varepsilon\).

      2. From the free energy, find expressions for the internal energy \(U\) and entropy \(S\) of the system.

      3. Plot the entropy versus \(T\). Explain its asymptotic behavior as the temperature becomes high.

      4. Plot the \(S(T)\) versus \(U(T)\). Explain the maximum value of the energy \(U\).

    • face Gibbs entropy approach

      face Lecture

      120 min.

      Gibbs entropy approach
      Thermal and Statistical Physics 2020

      Gibbs entropy information theory probability statistical mechanics

      These lecture notes for the first week of Thermal and Statistical Physics include a couple of small group activities in which students work with the Gibbs formulation of the entropy.
    • assignment Approximating a Delta Function with Isoceles Triangles

      assignment Homework

      Approximating a Delta Function with Isoceles Triangles
      Static Fields 2023 (6 years)

      Remember that the delta function is defined so that \[ \delta(x-a)= \begin{cases} 0, &x\ne a\\ \infty, & x=a \end{cases} \]

      Also: \[\int_{-\infty}^{\infty} \delta(x-a)\, dx =1\].

      1. Find a set of functions that approximate the delta function \(\delta(x-a)\) with a sequence of isosceles triangles \(\delta_{\epsilon}(x-a)\), centered at \(a\), that get narrower and taller as the parameter \(\epsilon\) approaches zero.
      2. Using the test function \(f(x)=3x^2\), find the value of \[\int_{-\infty}^{\infty} f(x)\delta_{\epsilon}(x-a)\, dx\] Then, show that the integral approaches \(f(a)\) in the limit that \(\epsilon \rightarrow 0\).

    • face Fermi and Bose gases

      face Lecture

      120 min.

      Fermi and Bose gases
      Thermal and Statistical Physics 2020

      Fermi level fermion boson Bose gas Bose-Einstein condensate ideal gas statistical mechanics phase transition

      These lecture notes from week 7 of Thermal and Statistical Physics apply the grand canonical ensemble to fermion and bosons ideal gasses. They include a few small group activities.
    • assignment Energy of a relativistic Fermi gas

      assignment Homework

      Energy of a relativistic Fermi gas
      Fermi gas Relativity Thermal and Statistical Physics 2020

      For electrons with an energy \(\varepsilon\gg mc^2\), where \(m\) is the mass of the electron, the energy is given by \(\varepsilon\approx pc\) where \(p\) is the momentum. For electrons in a cube of volume \(V=L^3\) the momentum takes the same values as for a non-relativistic particle in a box.

      1. Show that in this extreme relativistic limit the Fermi energy of a gas of \(N\) electrons is given by \begin{align} \varepsilon_F &= \hbar\pi c\left(\frac{3n}{\pi}\right)^{\frac13} \end{align} where \(n\equiv \frac{N}{V}\) is the number density.

      2. Show that the total energy of the ground state of the gas is \begin{align} U_0 &= \frac34 N\varepsilon_F \end{align}

    • keyboard Electrostatic potential of four point charges

      keyboard Computational Activity

      120 min.

      Electrostatic potential of four point charges
      Computational Physics Lab II 2023 (2 years)

      electrostatic potential python

      Students write python programs to compute and visualize the potential due to four point charges. For students with minimal programming ability and no python experience, this activity can be a good introduction to writing code in python using numpy and matplotlib.
    • face Review of Thermal Physics

      face Lecture

      30 min.

      Review of Thermal Physics
      Thermal and Statistical Physics 2020

      thermodynamics statistical mechanics

      These are notes, essentially the equation sheet, from the final review session for Thermal and Statistical Physics.
    • assignment Distribution function for double occupancy statistics

      assignment Homework

      Distribution function for double occupancy statistics
      Orbitals Distribution function Thermal and Statistical Physics 2020

      Let us imagine a new mechanics in which the allowed occupancies of an orbital are 0, 1, and 2. The values of the energy associated with these occupancies are assumed to be \(0\), \(\varepsilon\), and \(2\varepsilon\), respectively.

      1. Derive an expression for the ensemble average occupancy \(\langle N\rangle\), when the system composed of this orbital is in thermal and diffusive contact with a resevoir at temperature \(T\) and chemical potential \(\mu\).

      2. Return now to the usual quantum mechanics, and derive an expression for the ensemble average occupancy of an energy level which is doubly degenerate; that is, two orbitals have the identical energy \(\varepsilon\). If both orbitals are occupied the toal energy is \(2\varepsilon\). How does this differ from part (a)?

  • Thermal and Statistical Physics 2020 (3 years) Consider a three-state system with energies \((-\epsilon,0,\epsilon)\).
    1. At infinite temperature, what are the probabilities of the three states being occupied? What is the internal energy \(U\)? What is the entropy \(S\)?
    2. At very low temperature, what are the three probabilities?
    3. What are the three probabilities at zero temperature? What is the internal energy \(U\)? What is the entropy \(S\)?
    4. What happens to the probabilities if you allow the temperature to be negative?